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Aposyndetic continua as bundle spaces. (English) Zbl 0541.54040

For about ten years it was not known whether every aposyndetic, homogeneous curve \((=\) 1-dim continuum) was locally connected or not, the only known examples being the circle and the Menger universal curve. (The diadic solenoid is not aposyndetic.) Since the product of two nondegenerate continua is aposyndetic, it is easy to construct continua of higher dimension which are aposyndetic and homogeneous, but not locally connected. The first one dimensional one was constructed by Case using an inverse limit space generated in much the same way the diadic solenoid is generated by inverse limits. The author has discovered another and more natural way to generate these curves.
Let \({\mathcal S}=(S,f,S^ 1,G)\) be the P-adic solenoid bundle, where S is the P-adic solenoid, G is the P-adic integers, and \(f:S\to S^ 1\) is the projection of S into \(S^ 1\). If X is a continuum and \(\eta:X\to S^ 1\) is a map, then the bundle space B induced by the bundle \(\eta^{-1}{\mathcal S}\) is a nonlocally connected compactum of the same dimension as X. If \(\eta:X\to S^ 1\) is monotone, then B is a continuum. Or, if X is arcwise connected and contains a simple closed curve C such that \(\eta | C\) is a homeomorphism, then B is a continuum. If X is colocally arcwise connected and contains two disjoint simple closed curves such that \(\eta\) restricted to each is a homeomorphism, then B is an aposyndetic continuum; in fact, B is colocally connected. Or if X is locally peripherally connected and B is a continuum, then B is aposyndetic. Finally, if X is strongly locally homogeneous, then B is homogeneous. It is easy to see how to use these theorems to produce the desired curves. For example, if A is the boundary of the unit square from which is produced the universal plane curve and \(\eta\) is composed of the retraction of it onto A followed by a homeomorphim onto \(S^{-1}\), then the bundle space is an aposyndetic, nonlocally connected one-dimensional continuum. The author demonstrates how this machine can generate many other examples including the Case continuum.
Reviewer: F.B.Jones

MSC:

54F15 Continua and generalizations
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
54C56 Shape theory in general topology
55R10 Fiber bundles in algebraic topology
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References:

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